The present invention relates to the field of wireless cellular networks, more specifically to an approach for reducing interference of a terminal of such a wireless cellular network using interference alignment via minimizing projector distances of interfering subspaces.
FIG. 1 shows a schematic representation of a wireless cellular network comprising K cells, each cell including a respective base station BS serving one or more terminals or user equipment UE in the cell. FIG. 1 is a schematic representation and only one user equipment UE is shown in each cell, however, it is noted that a base station BS of a cell may serve a plurality of user equipment UE, like mobile phones. In FIG. 1 an exemplary case of a downlink transmission in a multi-cell MIMO/CoMP system (MIMO=Multiple Input Multiple Output; CoMP=Coordinated Multi-Point) is shown. The base stations BSs employ joint precoding to reduce/cancel inter-cell interference for the users UE, especially for cell-edge users UE1 to UEK as depicted in FIG. 1. Each base station provides for a direct link to the user equipment it serves, as is indicated by the solid line arrow. In addition, especially cell-edge users experience interference from other base stations as indicated by the dotted arrows in FIG. 1. For example, in cell 2 user or user equipment UE2 is located close to an edge of the cell, thereby also experiencing interference from the base station BS1 and the base station BSK in cells 1 and K, respectively. The base stations of the network or at least a number of the base stations are connected via a backhaul network having a central node CN.
FIG. 2 is a schematic representation of the channels or links between the respective base stations BSs and the user equipment UEs, wherein the direct links and the interfering links together with the associated channel matrices H are shown. Each base station uses a precoder F1 to FK, wherein by means of joint precoding, the inter-cell interference, especially for cell-edge users may be reduced or even canceled. At the user equipments, respective receive filters G1 to GK are used. For obtaining the reduction or cancelation of the inter-cell interference the precoders F1 to FK need to be designed such that the rate of the edge-users is maximized. Designing the precoders and the receive filters separately, in general, leads to a suboptimal solution and consequently to low user rates. Further, there are no easy analytical solutions due to the complexity of the problem. In known technology, some approaches for maximizing user rates in a wireless network are described by K. Gomadam, V. R. Cadambe and S. A. Jafar, “Approaching the Capacity of Wireless Networks through Distributed Interference Alignment”, in IEEE Global Telecommunications Conference (GLOBECOM), 2008. Two algorithms for maximizing user rates are described, the so-called interference leakage algorithm and the so-called Max-SINR algorithm. The goal of the interference leakage algorithm is to minimize interference power at the receivers which is obtained by a joint design of the precoding and receive matrices. While the performance obtained is good, i.e. the data rates obtained are good, the computational complexity is high and it is necessitated to have knowledge about the interfering links at the base stations. The Max-SINR algorithm has the goal to maximize the signal-to-interference-ratio at the receivers. Also, in this algorithm the precoding and receiving matrices are jointly designed. The performance, i.e. the obtainable data rates, is better when compared to the interference leakage algorithm, however, the computational complexity is higher and global channel knowledge needs to be available at the base stations, i.e. knowledge about the interfering links and the direct links.
Thus, with conventional approaches it is necessitated to design precoders and receive filters jointly which is a mathematically complex problem and, in addition, the receive filters need to be re-calculated every time the precoders change. Further, the obtained receive filters may have a complex structure (which is hard to calculate), and which cannot be employed in real scenarios/standards where MMSE or IRC filters (MMSE=minimum mean-square-error; IRC=interference-rejection-combining) are used. In the following, further details of the above described conventional approaches will be given.
When transmitters and receivers have multiple antennas, interference alignment (IA) approaches may be used. Interference alignment is based on the idea that unintended interference at each receiver can be forced to lie in only a subspace of the received signal; thus leaving another interference-free subspace which can be used for intended signal transmission. For example, if a receiver has two antennas, the unintended interference can be forced to lie in a one-dimensional subspace, leaving another subspace (also a one-dimensional subspace) to be interference-free. Interference alignment may be of specific interest when the antenna configuration of the system (the number of transmit and receive antennas) does not allow for a zero-forcing precoder design in accordance with which interference is pre-canceled at the transmitter side. In such a case IA offers an attractive alternative by first aligning the interference at each receiver by proper precoding at the transmitter side and then applying zero-forcing receive filters in order to cancel it.
In the following description, a system model will be considered having a K-user MIMO interference channel (IC) with K≧3 and where a user denotes a transmitter/receiver pair. This corresponds to K cells with K cell-edge users which are served using the same resource blocks; thus, each cell-edge user experiences interference from K−1 cells. Each transmitter is equipped with M antennas while each receiver is equipped with N antennas. A receiver desires to receive 1≦d<min(N, M) data symbols (streams). Prior to transmission, the data symbols sk˜NJ (O, Id) εCd are precoded at a transmitter k by the linear precoder FkεCM×d, are sent over the direct channel HkkεCN×M, and are received by receiver k. Each transmitter k has a transmit power constraint E[∥Fksk∥22]=tr(FkHFk)=Etx,l. At the receiver k the obtained signal is perturbed by noise nk˜NC(O,Cnk)εCN in addition to unintended interference coming from other transmitters l≠k. A receive filter GkεCN×d is employed to reduce or cancel the interference and/or maximize the achievable rate. The estimates of the data symbols received at the receiver k via the direct channel from transmitter k are given as follows:
                                          s            ^                    k                =                              G            k            H                    ⁡                      (                                                            H                  kk                                ⁢                                  F                  k                                ⁢                                  s                  k                                            +                                                ∑                                                            l                      =                      1                                        ,                                          l                      ≠                      k                                                        K                                ⁢                                                                  ⁢                                                      H                    kl                                    ⁢                                      F                    l                                    ⁢                                      s                    l                                                              +                              n                k                                      )                                              (        1        )            where HklεC N×M denotes the channel between the receiver k and the transmitter l. Conventionally, it is assumed that rank (Hkl)=min(N, M), ∀k, l. In the present case, this constraint is relaxed to (Hkl)≧d.
The total achievable rate of all users R is given by:
      R    =                  ∑                  k          =          1                K            ⁢                          ⁢              R        k              ,where Rk is the achievable rate of user k. Equation (1) may be rewritten as follows:
                                                        s              ^                        k                    =                                                                      H                  ^                                k                            ⁢                              s                k                                      +                                          ∑                                                      l                    =                    1                                    ,                                      l                    ≠                    k                                                  K                            ⁢                                                          ⁢                                                                    H                    ^                                                        int                    ,                    l                                                  ⁢                                  s                  l                                                      +                                          G                k                H                            ⁢                              n                k                                                    ,                            (        2        )            where Ĥk=GkHHkkFk and Ĥint,l=GkHHklFl denote the effective direct channel between the k-th transmitter/receiver pair and the effective interfering channel between the transmitter l and receiver k, respectively. Rk can then be written as follows:
                              R          k                =                              log            2                    ⁢                      det            ⁡                          (                                                I                  d                                +                                                                            H                      ^                                        k                                    ⁢                                                                                                              H                          ^                                                k                        H                                            ⁡                                              (                                                                                                            ∑                                                                                                l                                  =                                  1                                                                ,                                                                  l                                  ≠                                  k                                                                                            K                                                        ⁢                                                                                                                  ⁢                                                                                                                            H                                  ^                                                                                                  int                                  ,                                  l                                                                                            ⁢                                                                                                H                                  ^                                                                                                  int                                  ,                                  k                                                                H                                                                                                              +                                                                                    G                              k                              H                                                        ⁢                                                          C                                                              n                                k                                                                                      ⁢                                                          G                              k                                                                                                      )                                                                                    -                      1                                                                                  )                                                          (        3        )            
In order to achieve interference alignment (IA) the following two conditions has to hold:GkHHklFl=0, ∀l≠k rank(GkHHkkFk)=d.   (4)
The first condition simply states that interference from all unintended transmitters should be suppressed, while the second one states that the design of the precoders and the receive filters needs to ensure the existence of an effective interference-free channel between the k-th transmitter/receiver pair where d data symbols (streams) can be simultaneously communicated.
The above-mentioned two conventional methods/algorithms for maximizing user rates in a wireless network will now be described in further detail. Both methods are based on the concept of network reciprocity, a concept which holds for Time Division Duplex (TDD) based systems. Due to the reciprocity, the signaling dimensions along which a receiving node sees the least interference from transmit signals are also the same dimensions along which the node will cause the least interference to other nodes in the reciprocal network where the roles of transmitters and receivers are switched. Moreover, the concept of reciprocity implies that the channel between the receiver l and the transmitter k in the reciprocal network Hlkr is related to the channel between the receiver k and the transmitter l in the original network as follows:Hlkr=HklH   (5)
Additionally, when defining Fkr and Gkr to be the precoding and receive filters of user k in the reciprocal network. The IA conditions can be written as follows:Glr,HHlkrFkr=0, ∀k≠l rank(Gkr,HHkkrFkr)=d.   (6)
As can be seen, when setting Fkr=Gk and Gkr=Fk, equation (6) becomes equivalent to equation (4) which tells that the alignment is reciprocal. Thus, alignment in the reciprocal networks can be achieved if it can be achieved in the original network, and interference alignment in the reciprocal network can be achieved by choosing the precoding and receive filters of the reciprocal network to be the receive and precoding filters of the original network.
In accordance with this interference alignment method the signals are not separated in space; rather, at each step of the algorithm, it is tried to minimize the interference leakage (power) at each receiver such that the first condition of equation (4) is fulfilled when the algorithm converges. If interference power is zero, then interference coming from undesired transmitters is implicitly aligned to a smaller dimension. The interference leakage Ik of receiver k in the original network due to all transmitters l≠k is defined as:
                                          I            k                    =                      tr            ⁡                          [                                                G                  k                  H                                ⁢                                  Q                  k                                ⁢                                  G                  k                                            ]                                      ,                                  ⁢        where                            (        7        )                                          Q          k                =                              ∑                                          l                =                1                            ,                              l                ≠                k                                      K                    ⁢                                          ⁢                                                    E                                  tx                  ,                  l                                            d                        ⁢                          H              kl                        ⁢                          F              l                        ⁢                          F              l              H                        ⁢                          H              kl              H                                                          (        8        )            is the interference covariance matrix of receiver k. For given precoders, the columns of the receive filter that minimizes Ik are the eigenvectors corresponding to the d smallest eigenvalues of Qk, that is:Gk:,l=vl(Qk),   (9)
where vl(Qk) is the eigenvector corresponding to the l-th smallest eigenvalue of Qk and Gk:,l is the l-th column of Gk. In the reciprocal network, the interference leakage Ikr is defined as follows:
                                                        I              k              r                        =                          tr              ⁡                              [                                                      G                    k                                          r                      ,                      H                                                        ⁢                                      Q                    k                    r                                    ⁢                                      G                    k                    r                                                  ]                                              ,                                          ⁢          where                ⁢                                  ⁢                              Q            k            r                    =                                                    ∑                                                      l                    =                    1                                    ,                                      l                    ≠                    k                                                  K                            ⁢                                                          ⁢                                                                    E                                          tx                      ,                      l                                                        d                                ⁢                                  H                  kl                  r                                ⁢                                  F                  l                  r                                ⁢                                  F                  l                                      r                    ,                    H                                                  ⁢                                  H                  kl                                      r                    ,                    H                                                                        =                                          ∑                                                      l                    =                    1                                    ,                                      l                    ≠                    k                                                  K                            ⁢                                                          ⁢                                                                    E                                          tx                      ,                      l                                                        d                                ⁢                                  H                  lk                  H                                ⁢                                  G                  l                                ⁢                                  G                  l                  H                                ⁢                                  H                  lk                                                                                        (        10        )            is the interference covariance matrix of receiver k. Similarly, the columns of the receive filter that minimizes Ikr are the eigenvectors corresponding to the d smallest eigenvalues of Qkr.
The algorithm alternates between the original and reciprocal networks. Within each network the receivers update their receive filters such that the interference leakage is minimized. Arbitrary initial orthonormal precoders may be chosen as a starting point, and details of the algorithm for interference alignment via minimizing interference leakage are depicted in FIG. 3.
For implementing this algorithm, the channel state information (CSI) between receivers and undesired transmitters need to be available (i.e., the quantities Hkl, ∀k, ∀l≠k). The CSI of the direct links is not needed. To achieve IA, an alternating optimization over both the precoding and receive filters is performed; with the precoders fixed, the receive filters are updated, and vice versa. An implementation of the algorithm in a distributed fashion can be realized as follows:
(1) To update the receive filters, the precoders are exchanged between the transmitters and channel matrices of the original network are used in the optimization process.
(2) To update the precoders, the receive filters are exchanged between the transmitters and the channel matrices of the reciprocal network are used in the optimization process.
For this approach it is assumed that the transmitters are responsible for calculating the receive filters which will be signaled to the receivers after the algorithm converges.
The second algorithm, the Max-SINR algorithm, aims at directly maximizing the Signal-to-Interference-Noise (SINR) ratio of each desired transmitted stream. When compared to the first algorithm, which tries to perfectly align interference in a lower dimensional subspace, the Max-SINR algorithm tries to maximize the desired signal power within the desired signal space. The SINR of the j-the stream of the k-the receiver in the original network is defined as follows:
                              SINR                      k            ,            j                          =                                                            G                k                                  :                                      ,                    j                    ,                    H                                                              ⁢                              H                kk                            ⁢                              F                k                                  :                                      ,                    j                                                              ⁢                              F                k                                  :                                      ,                    j                    ,                    H                                                              ⁢                              H                kk                H                            ⁢                              G                k                                  :                                      ,                    j                                                                                                      G                k                                  :                                      ,                    j                    ,                    H                                                              ⁢                              B                                  k                  ,                  j                                            ⁢                              G                k                                  :                                      ,                    j                                                                                ⁢                                    E                              tx                ,                k                                      d                                              (        11        )            where Fk:,j is the j-th column of Fk and Bk,j is the interference plus noise covariance matrix of the j-th stream of the k-th receiver:
                              B                      k            ,            j                          =                                            ∑                              l                =                1                            K                        ⁢                                                  ⁢                                                            E                                      tx                    ,                    l                                                  d                            ⁢                                                ∑                                      u                    =                    1                                    d                                ⁢                                                                  ⁢                                                      H                    kl                                    ⁢                                      F                    l                                          :                                              ,                        u                                                                              ⁢                                      F                    l                                          :                                              ,                        u                        ,                        H                                                                              ⁢                                      H                    kl                    H                                                                                -                                                    E                                  tx                  ,                  k                                            d                        ⁢                          H              kk                        ⁢                          F              k                              :                                  ,                  j                                                      ⁢                          F              k                              :                                  ,                  j                  ,                  H                                                      ⁢                          H              kk              H                                +                      C                          n              k                                                          (        12        )            
With this definition, the unit vector Gk:,j that maximizes SINRk,j is given by:
                              G          k                      :                          ,              j                                      =                                                            B                                  k                  ,                  j                                                  -                  1                                            ⁢                              H                kk                            ⁢                              F                k                                  :                                      ,                    j                                                                                                                                                            B                                          k                      ,                      j                                                              -                      1                                                        ⁢                                      H                    kk                                    ⁢                                      F                    k                                          :                                              ,                        j                                                                                                                        2                                .                                    (        13        )            
A similar analysis is done in the reciprocal network; for brevity, it is skipped and the Max-SINR algorithm is shown in further detail in FIG. 4.
This algorithm necessitates global channel knowledge to be implemented, i.e., every transmitter should know the CSI of both the direct links and the interfering links. When compared to the first algorithm, this is a big signaling overhead. For example, for a system of K=3 transmitter/receiver pairs, this constitutes a 50% additional overhead (9 channels need to be communicated from the receivers to the transmitters in total instead of 6 channels in the first algorithm). This algorithm can be implemented in a distributed fashion where the precoders and receive filters are iteratively exchanged between the transmitters, similar to the first algorithm. Similar to the first algorithm, an alternating optimization over both the precoding and receive filters may be performed.
The above description shows that the conventional approaches are disadvantageous because the precoders and receive filters are jointly designed resulting in the necessity to solve mathematically complex problems and, further, receive filters need to be re-calculated every time a precoder changes. Further, the receive filters will have a complex structure which is hard to calculate so that they cannot easily be used in real scenarios/standards.